For the sake of completeness, in this section we present the leave-one-out analysis due to [1]}. Following the notation used in the proof of Theorem REF , we introduce an additional independent element \((X_{n + 1}, Y_{n + 1})\)  distributed according to \(P\)  to the sample \(S_{n+1}\) . With a slight abuse of notation, we define for \(j = 1, \ldots , n + 1\)  the unnormalized and unpenalized (as opposed to (REF )) empirical second moment matrices by
\(\tilde{\Sigma } = \sum \limits _{i = 1}^{n + 1}X_iX_i^{\mathsf {T}} \quad \text{and} \quad \tilde{\Sigma }^{(-j)} = \sum \limits _{i = 1, i \ne j}^{n + 1}X_iX_i^{\mathsf {T}}.\) 
